Calculus displacement?

Question

The displacement, y centimeters, of the slide valve of an engine is given by the expression.

y= 2.6 cos 5π t + 3.8 sin 5π t.

what is the expression for the velocity, v, of the valve
and what is the velocity in m/s when t = 20m/s.

Answer 1

v = dy/dt = -13pi sin(5pi t) + 19pi cos(5pi t)

when t = 20s,
v = -13pi sin(100pi) + 19pi cos(100 pi)
= -13pi(0) + 19pi(1) = 19pi

Answer 2

The velocity of the slide valve is given by the rate of change of displacement, that is dy/dt. So the first step is to differentiate this expression with respect to t to obtain an expression for the velocity of the valve.

dy/dt= -2.5*5pi*sin(5pi*t)+3.8*5pi*cos(5pi*t)

dy/dt= -13pi*sin(5pi*t)+19pi*cos(5pi*t)

To obtain the velocity when t=20 s, substitute 20 for t in this expression:

dy/dt= -13pi*sin(5pi*20)+19pi*cos(5pi*20)

Then evaluate remembering to have your calculator in radian mode:

dy/dt = -13pi*0+19pi*1

dy/dt = 19pi m/s (exact answer)

or dy/dt ~ 59.69026 m/s , ~ 59.7 m/s (3 sig.fig)

Remember as well that velocity is a vector so for a complete answer you need to specify the direction not just the magnitude. 'In the direction the slide valve is moving' is sufficient.

Sorry, for some reason the ends of some of the expression are cut off but I am sure you fill in the missing bits!

Answer 3

v1= 5+15*3= 50 m/s d1=50*3=150 m v2= 5+15*6=95 m/s d2= 95*6= 570m d=d2-d1= 420m

Answer 4

f `(t)
= (- 5π).(2.6).sin (5πt) + (5π).(3.8).cos (5πt)
= (- 40.8).sin (5πt)+ 59.7.cos (5πt)

f `(20)
= (- 40.8).sin(100π) + 59.7.cos(100π)
= 5.71 + 0.99
= 6.7

velocity = 6.7 m/s

Answer 5

Vy = - 13πsin(5πt) + 19πcos(5πt)
Vy(20 ms) = - 13πsin(0.1π) + 19πcos(0.1π)
Vy(20 ms) = 44.148 m/s

Answer 6

vel=y'=-13Pisin5Pi*t+19Picos5Pi*t..subst t=20s..
>y'=19Pi..or about 60m/s in mag..